If the value `int(1-(cotx)^(2008))/(tanx+(cotx)^(2009))dx=1/k ln|sin^kx+cos^kx|+c`,then find k.

Explanation

To solve the integral

\[ I = \int \frac{1 - (\cot x)^{2008}}{\tan x + (\cot x)^{2009}} \, dx, \]

we will first rewrite the expression in terms of sine and cosine.

### Step 1: Rewrite in terms of sine and cosine

Recall that:
- \(\tan x = \frac{\sin x}{\cos x}\)
- \(\cot x = \frac{\cos x}{\sin x}\)

Substituting these into the integral gives:

\[ I = \int \frac{1 - \left(\frac{\cos x}{\sin x}\right)^{2008}}{\frac{\sin x}{\cos x} + \left(\frac{\cos x}{\sin x}\right)^{2009}} \, dx. \]

This simplifies to:

\[ I = \int \frac{1 - \frac{\cos^{2008} x}{\sin^{2008} x}}{\frac{\sin^2 x}{\cos x} + \frac{\cos^{2009} x}{\sin^{2009} x}} \, dx. \]

### Step 2: Simplify the numerator and denominator

The numerator becomes:

\[ \frac{\sin^{2008} x - \cos^{2008} x}{\sin^{2008} x}. \]

The denominator can be expressed as:

\[ \frac{\sin^2 x \cdot \sin^{2009} x + \cos^{2010} x}{\sin^{2009} x \cdot \cos x}. \]

Thus, we have:

\[ I = \int \frac{\sin^{2008} x - \cos^{2008} x}{\sin^{2008} x} \cdot \frac{\sin^{2009} x \cdot \cos x}{\sin^2 x \cdot \sin^{2009} x + \cos^{2010} x} \, dx. \]

### Step 3: Combine and simplify

Taking the LCM in the denominator, we can rewrite:

\[ I = \int \frac{\sin^{2008} x - \cos^{2008} x}{\sin^{2010} x + \cos^{2010} x} \, dx. \]

### Step 4: Substitution

Let \( t = \sin^{2010} x + \cos^{2010} x \). Then, differentiating both sides gives:

\[ dt = 2010 \left( \sin^{2009} x \cos x \, dx - \cos^{2009} x \sin x \, dx \right). \]

Rearranging gives:

\[ \frac{dt}{2010} = \sin^{2009} x \cos x \, dx - \cos^{2009} x \sin x \, dx. \]

### Step 5: Substitute back into the integral

Substituting back into the integral, we have:

\[ I = \frac{1}{2010} \int \frac{dt}{t}. \]

This integral evaluates to:

\[ I = \frac{1}{2010} \ln |t| + C = \frac{1}{2010} \ln |\sin^{2010} x + \cos^{2010} x| + C. \]

### Step 6: Compare with given expression

We are given that:

\[ I = \frac{1}{k} \ln |\sin^k x + \cos^k x| + C. \]

From our result, we can see that \( k = 2010 \).

### Final Answer

Thus, the value of \( k \) is:

\[ \boxed{2010}. \]